Optimum construction of a private network-to-network interface

ABSTRACT

The number of nodes in a private network-to-network interface may be very large. These nodes can be organized in two or more hierarchical levels. 
     Network performance parameters, for example, call set-up time, may be used to determine the optimum number of hierarchical levels for a given PNNI network. A multi-step algorithm is presented for calculating the optimum number of levels in the network for a given number of nodes.

FIELD OF THE INVENTION

This invention relates to private network-to-network interfaces (PNNI),and more specifically to methods for constructing PNNI networks withoptimized architecture.

BACKGROUND OF THE INVENTION

Private network-to-network interfaces (PNNI) refers to very large ATMnetworks that communicate between multiple private networks. Dependingon the number of nodes served by the PNNI network, the architecture maybe flat (non-hierarchical) or may have two or more hierarchical levels.When the network topology moves to multiple level architecture, severalnew effects on network performance are introduced. A primary motive forintroducing new levels in a PNNI network is to increase routingefficiency and reduce call set-up time. However, added levels increasecost and management complexity. Thus there is an important trade-offwhen considering adding new hierarchical levels.

A typical PNNI network is organized in peer groups. These are subunitsof the total network that are interconnected as a group, with each groupthen interconnected in the PNNI network. An advantage of thisarchitecture is that communications between members in a peer group canbe independent of overall network management. When a communicationbetween a node in one peer group and a node in another peer group isinitiated, the PNNI protocol is used for that call set-up andmanagement. The size of the peer groups, i.e. the number of nodes ineach peer group, also affects the overall performance of the PNNInetwork. Methods for optimizing peer group size are described andclaimed in my co-pending application Ser. No. 10/918,913, filed Aug. 16,2004.

To make an accurate evaluation of the trade-off described above requiresthat the changes in call set-up times when a new hierarchical level isadded be known with a high confidence level.

BRIEF STATEMENT OF THE INVENTION

According to the invention, a technique has been developed thatevaluates important network performance parameters, for example, callset-up time, for different network topologies. These are used todetermine the optimum number of hierarchical levels for a given PNNInetwork.

BRIEF DESCRIPTION OF THE DRAWING

The invention may be better understood when considered in conjunctionwith the drawing in which:

FIG. 1 is a schematic diagram of a flat (non-hierarchical) PNNI network;

FIG. 2 is a schematic representation of a two-level hierarchical PNNInetwork.

DETAILED DESCRIPTION OF THE INVENTION

With reference to FIG. 1, a flat topological PNNI network isrepresented.

In the discussion that follows, let N be the total number oflowest-level nodes (i.e., the number of ATM switches). Let x₁ be thenumber of lowest-level nodes in each level-1 PG, and let x₂=N/x₁ be thenumber of level-1 PGs (all variables are assumed to be continuous).

Define A={α|α>1}. We assume the time complexity of computing a minimumcost path in a flat (non-hierarchical) network with z nodes isR₁(z)=α₀z^(α), where α₀>0 and αεA. For example, for Dijkstra's shortestpath method we have α=2.

Certain nodes are identified as border nodes. A level-1 border node of aPG is a lowest-level node which is an endpoint of a trunk linking the PGto another level-1 PG. For example, if each U.S. state is a level-1 PG,and if there is a trunk from switch a in Chicago to switch b in Denver,then a and b are level-1 border nodes. Define Γ={γ|0≦γ<1}. We assume thenumber of level-1 border nodes in a PG with x₁ lowest-level nodes isbounded above by B₁(x₁)=γ₀x₁ ^(γ), where γ₀>0 and γεΓ. The case whereeach PG has a constant number of border nodes is modelled by choosingγ₀=k and γ=0. The case where the border nodes are the (approximately4√{square root over (x₁)}) boundary nodes of a square grid of x₁switches is modelled by choosing γ₀=4 and γ=½.

Define K={κ|0≦κ≦1}. We assume the total number of level-1 PGs, excludingthe source level-1 PG, that the connection visits is bounded above byV₁(x₂)=κ₀(x₂ ^(κ)−1), where κ₀>0 and κεK. Note that V₁(1)=0, which meansthat in the degenerate case of exactly one level-1 PG, there are no PGsvisited other than the source PG.

This functional form for V₁(x₂) is chosen since the expected distancebetween two randomly chosen points in a square with side length L is kL,where k depends on the probability model used: using rectilineardistance, with points uniformly distributed over the square, we havek=⅔; using Euclidean distance we have k=( 1/15)[2+√{square root over(2)}+5 log(√{square root over (2)}+1)]≈0.521; with an isotropicprobability model we have k=[(2√{square root over (2)}/(3π)]log(1+√{square root over (2)})≈0.264. If the x₂ PGs are arranged in asquare grid, a random connection will visit approximately k√{square rootover (x₂)} PGs. Choosing κ₀=k and κ=½, the total number of PGs visitedis approximately κ₀x₂ ^(κ). Choosing κ₀=1 and κ=1 models the worst casein which the path visits each level-1 PG.

The source node of a connection in a two-level network sees the x₁ nodesin the source PG, and at most B₁(x₁) border nodes in each of the x₂−1non-source PGs. A path computation is performed by the entry border nodeof at most V₁(x₂) non-source PGs. In each non-source PG visited, theentry border node sees only the x₁ nodes in its PG when computing a pathacross the PG (or a path to the destination node, in the case of thedestination PG), and so the path computation time complexity at eachentry border node is R₁(x₁). Hence the total path computation time isbounded above by R₁(x₁+(x₂−1)B₁(x₁))+V₁(x₂)R₁(x₁)=α₀[x₁+(x₂−1)γ₀x₁^(γ)]^(α)+κ₀(x₂ ^(κ)−1)α₀x₁ ^(α). We ignore the constant factor α₀. Theoptimization problem for a 2-level hierarchy is thus: minimize[x₁+(x₂−1)γ₀x₁ ^(γ)]^(α)+κ₀x₁ ^(α)(x₂ ^(κ)−1) subject to x₁x₂=N.

We next transform this optimization problem to a convex optimizationproblem (which has a convex objection function, a convex feasibleregion, and any local minimum is also a global minimum. We approximatex₂−1 by x₂ and x₂ ^(κ)−1 by x₂ ^(κ), yielding the objective function[x₁+γ₀x₁ ^(γ)x₂]^(α)+κ₀x₁ ^(α)x₂ ^(κ), which also upper bounds the totalpath computation time. We rewrite the constraint x₁x₂=N as Nx₁ ⁻¹x₂⁻¹=1, which can be replaced by Nx₁ ⁻¹x₂ ⁻¹≦1, since the inequality mustbe satisfied as an equality at any solution of the optimization problem.Letting y=x₁+γ₀x₁ ^(γ)x₂ yields the optimization problem: minimizeγ^(α)+κ₀x₁ ^(α)x₂ ^(κ) subject to x₁+γ₀x₁ ^(γ)x₂≦y and Nx₁ ⁻¹x₂ ⁻¹≦1.The inequality constraint x₁+γ₀x₁ ^(γ)x₂≦y must be satisfied as anequality in any solution; we rewrite this constraint as x₁y⁻¹+γ₀x₁^(γ)x₂y⁻¹≦1. Let s₁=log x₁, s₂=log x₂, s=(s₁, s₂), and t=log y.Combining exponential terms, we obtain the optimization problem T₂(N):minimize ƒ₂(s,t)=e^(αt)+κ₀e^((αs) ¹ ^(+κs) ² ⁾  (1)subject to e^((s) ¹ ^(−t))+γ₀e^((γs) ¹ ^(+s) ² ^(−t))≦1;  (2)Ne^((−s) ¹ ^(−s) ² ⁾≦1.  (3)Problem T₂(N) it is a special type of convex optimization problem calleda geometric program. Geometric programs are particularly well suited toengineering design, with a rich duality theory permitting particularlyefficient solution methods.

Let ƒ₂*(N) be the infimum of the primal objective function (1) subjectto the primal constraints (2) and (3). Thus ƒ₂*(N) is the minimum totalpath computation time for a two-level PNNI network with N lowest-levelnodes.

Define

$\begin{matrix}{\omega = {{\alpha\;\gamma} + {\frac{{\alpha^{2}\left( {1 - \gamma} \right)}^{2}}{{\alpha\left( {2 - \gamma} \right)} - \kappa}\mspace{14mu}{and}}}} & (4) \\{{\omega_{0} = {\alpha^{\alpha}{\mathbb{e}}^{{({2 + \kappa_{0} + \gamma_{0}})}/e}}},} & (5)\end{matrix}$Theorem 1 below states that ω₀N^(ω) is an upper bound on the minimumtotal path computation time for two-level PNNI networks.Theorem 1. For all N>0 we have ƒ₂*(N)≦ω₀N^(ω).

By assumption, the computational complexity of routing in a flat networkwith N nodes has complexity α₀z^(α). That is, flat network routing hascomputational complexity O(N^(α)). (Recall that a function g(z) of thesingle variable z is O(x^(η)) if g(z)≦η₀z^(η) for some η₀>0 and allsufficiently large z.) Theorem 1 above states that the computationalcomplexity of routing in a two-level network with N nodes is O(N^(ω)).

Theorem 2. ω<α.

Theorem 2 implies that, for all sufficiently large N, the computationalcomplexity of path computation is smaller for a two-level network thatfor a flat network. By evaluating ω, the advantage of using a two-levelnetwork, rather than a flat network, can be computed. The larger thedifference α−ω, the larger the computational savings of using atwo-level network. With α=2 and κ=γ=½, we have ω=7/5, so the minimumtotal path computation time is O(N^(7/5)), rather than the flat networkvalue O(N²). With α=2, κ=½ and γ=0, we have ω=8/7, so having a constantnumber of border nodes for each PG reduces the total routing time toO(N^(8/7)).

For a three-level network, let N be the total number of lowest-levelnodes, let x₁ be the number of lowest-level nodes in each level-1. PC,x₂ be the number of level-1. PCs in each level-2 PG, and x₃ be thenumber of level-2 PGs. Thus x₁x₂x₃=N.

As for H=2, we assume the complexity of routing in a flat network with zlowest-level nodes is R₁(z)=α₀z^(α), where α₀>0 and αεA.

As for H=2, certain nodes are identified as border nodes. A level-1border node of a PG in a 3-level network is a lowest-level node which isan endpoint of a trunk linking the PG to another level-1 PG within thesame level-2 PG. A level-2 border node of a PG in a 3-level network is alowest-level node which is an endpoint of a trunk linking the PG toanother level-2 PG within the same PNNI network. For example, supposeeach country in the world is a level-2 PG, and each U.S. state is alevel-1 PG. Then if there is a trunk from a switch a in Boston to aswitch b in London, a and b are level-2 border nodes.

For h=1, 2, we assume that the number of level-h border nodes in alevel-h PG with z lowest-level nodes is bounded above byB_(h)(z)=γ₀z^(γ), where γ₀>0 and γεΓ. Thus each level-1 PG has at mostB₁(x₁)=γ₀x₁ ^(γ) level-1 border nodes, and each level-2 PG has at mostB₂(x₁x₂)=γ₀(x₁x₂)^(γ) level-2 border nodes.

We assume that the total number of level-2 PGs, excluding the sourcelevel-2 PG, that the connection visits is bounded above by V₂(x₃)=κ₀(x₃^(κ)−1), where κ₀>0 and κεK. Note that V₂(1)=0, which means that in thedegenerate case where there is one level-2 PG, there are no level-2 PGsvisited other than the source level-2 PG. We assume that the totalnumber of level-1 PGs visited within the source level-2 PG, excludingthe source level-1 PG, is bounded above by V₁(x₂)=κ₀(x₂ ^(κ)−1).

In a three-level PNNI network, the source node sees the x₁ nodes in itslevel-1 PG, at most B₁(x₁) level-1 border nodes in each of the x₂−1level-1 PGs (excluding the source level-1 PG) in the same level-2 PG asthe source, and at most B₂(x₁x₂) level-2 border nodes in each of thex₃−1 level-2 PGs (excluding the source level-2 PG) in the PNNI network.Thus the total number of nodes seen by the source is bounded above byx ₁+(x ₂−1)B ₁(x ₁)+(x ₃−1)B ₂(x ₁ x ₂)=x ₁+(x ₂−1)γ₀ x ₁ ^(γ)+(x₃−1)γ₀(x ₁ x ₂)^(γ).The time complexity of the source path computation is bounded above byR ₁(x ₁+(x ₂−1)B ₁(x ₁)+(x ₃−1)B ₂(x ₁ x ₂)).

The total path computation time for all the level-1 PGs in the sourcelevel-2 PG, excluding the source level-1 PG, is at most V₁(x₂)R₁(x₁).For z>0, define R₂(z)=ω₀z^(ω). The total path computation time for eachof the V₂(x₃) level-2 PGs visited (other than the source level-2 PG) is,by definition, ƒ₂*(x₁x₂), which by Theorem 2 is bounded above byR₂(x₁x₂).

To minimize the upper bound on the total path computation time for athree-level network, we solve the optimization problem:minimize R₁(x ₁+(x ₂−1)B ₁(x ₁)+(x ₃−1)B ₂(x ₁ x ₂))+V ₁(x ₂)R ₁(x ₁)+V₂(x ₃)R ₂(x ₁ x ₂)subject to x₁x₂x₃=N. We approximate x₂−1 by x₂, x₃−1 by x₃, x₂ ^(κ)−1 byx₂ ^(κ), and x₃ ^(κ)−1 by x₃ ^(κ), which preserves the upper bound.Introducing the variable y, we obtain the optimization problem: minimizeα₀y^(α)+κ₀x₂ ^(κ)α₀x₁ ^(α)+κ₀x₃ ^(κ)ω₀(x₁x₂)^(ω) subject to x₁+x₂γ₀x₁^(γ)+x₃γ₀(x₁x₂)^(γ)≦y and Nx₁ ⁻¹x₂ ⁻¹x₃ ⁻¹≦1. Letting s₁=log x₁, s₂=logx₂, s₃=log x₃, s=(s₁, s₂, s₃), and t=log y, we obtain the geometricprogram T₃(N):minimize ƒ₃(s,t)=α₀e^(αt)+α₀κ₀e^((αs) ¹ ^(+κs) ² ⁾+κ₀ω₀e^((ω(s) ₁ ^(+s)₂ ^()+κs) ³ ⁾  (6)subject to e^(s) ¹ ^(−t)+γ₀e^((γs) ¹ ^(+s) ² ^(−t))+γ₀e^((γ(s) ¹ ^(+s) ²^()+s) ³ ^(−t))≦1  (7)Ne^((−s) ¹ ^(−s) ² ^(−s) ³ ⁾≦1  (8)

Let ƒ₃*(N) be the infimum of the primal objective function (6) subjectto constraints (7) and (8). Thus ƒ₃*(N) is the minimum total pathcomputation time for a three-level PNNI network with N lowest-levelnodes.

Define

$\begin{matrix}{\xi = {\frac{\alpha\left( {\omega - {\kappa\;\gamma}} \right)}{{\alpha\left( {1 - \gamma} \right)} + \left( {\omega - \kappa} \right)}.}} & (9)\end{matrix}$Theorem 3. For some ξ₀>0 and all N>0 we have ƒ₃*(N)≦ξ₀N^(ξ).

By Theorem 3, the minimum total path computation time for a three-levelPNNI network has computational complexity O(N^(ξ)).

Theorem 4.

${\omega - \xi} = {\frac{\left( {\omega - \kappa} \right)\left( {\omega - {\alpha\;\gamma}} \right)}{{\alpha\left( {1 - \gamma} \right)} + \left( {\omega - \kappa} \right)} > 0.}$

Theorem 4 implies that, for all sufficiently large N, the computationalcomplexity of path computation is smaller for a three-level PNNI networkthat for a two-level network. By evaluating ω and ξ, the advantage ofusing a three-level network, rather than a two-level network, can becomputed. The larger the difference ω−ξ, the larger the computationalsavings of using a three-level network. With α=2 and κ=γ=½, we haveω=7/5 and ξ=23/19≈1.21, so for three-level networks, the total pathcomputation time is O(N^(23/19)), rather than O(N^(7/5)) as would beobtained with a two-level network. With α=2, κ=½ and γ=0, we have ω=8/7and ξ= 32/37≈0.865, so the total path computation time is O(N^(32/37)),rather than O(N^(8/7)) as would be obtained with a two-level network; inthis case the total path computation time is sublinear in N for athree-level network.

The following steps determine whether to design the PNNI network as aflat (non-hierarchical) network, as a two-level network, or as athree-level network.

Step 1. Compute

$\omega = {{\alpha\;\gamma} + \frac{{\alpha^{2}\left( {1 - \gamma} \right)}^{2}}{{\alpha\left( {2 - \gamma} \right)} - \kappa}}$Step 2. Compute

$\xi = {\frac{\alpha\left( {\omega - {\kappa\;\gamma}} \right)}{{\alpha\left( {1 - \gamma} \right)} + \left( {\omega - \kappa} \right)}.}$Step 3. (a) If α−ξ≦τ₁, where τ₁ is a predetermined positive tolerance,then design the PNNI network as a flat (non-hierarchical) network. Ifα−ξ>τ₁, go to (b). (b) If ω−ξ≦τ₂, where τ₂ is a predetermined positivetolerance, then design the PNNI network as a two-level network. Ifω−ξ>τ₂, then design the PNNI network as a three-level network.

Various additional modifications of this invention will occur to thoseskilled in the art. All deviations from the specific teachings of thisspecification that basically rely on the principles and theirequivalents through which the art has been advanced are properlyconsidered within the scope of the invention as described and claimed.

1. A method for constructing a private network-to-network interface(PNNI) network, the PNNI network configured in PNNI network peer groupsubunits each comprising a plurality of nodes, comprising the steps of:a. assembling a first level of network nodes in PNNI network peer groupsby interconnecting nodes within each peer group, and interconnecting atleast some of the nodes of one peer group with nodes of another peergroup, so that a node in a peer group in the first level may beconnected to a node in another peer group of the first level through nintermediate nodes, b. adding a second level of network nodes whereinthe second level of network nodes are interconnected together andwherein the nodes of the second level provide interconnections betweenthe peer groups in the first level, so that at least some of the nodesin a peer group in the first level may be connected to a node in anotherpeer group of the first level through n−x intermediate nodes, where x isat least 1, wherein the step of adding the second level of network nodesis performed using the calculation: where, α represents the timecomplexity of computing a minimum cost path in a flat network, γ is aconstant characterizing the number of border nodes in the network, κcharacterizes the number of lower level peer groups in each higher levelpeer group that is visited by a network connection, τ is a predeterminedpositive tolerance, and ω, ξ, are temporary variables.